Conductors and Insulators — Question 1

Question 1

Successive contacts between identical spheres retain a memory of their order

Three identical conducting spheres on insulating supports initially carry $(Q_A,Q_B,Q_C)=(12,0,-4)$ nC. Each selected pair is connected by a thin wire, allowed to equilibrate in a symmetric arrangement far from the third sphere, and disconnected. Neglect wire charge; identical connected spheres then share their combined charge equally.

1. Find the final charges if A contacts B first, followed by B contacting C.

2. Repeat in reverse order and compare the outcomes and total charge.

Question 1 - Solution

Successive contacts between identical spheres retain a memory of their order

(a) Method and calculation

$$AB:\quad(12,0,-4)\longrightarrow(6,6,-4)\,\mathrm{nC}.$$$$BC:\quad(6,6,-4)\longrightarrow(6,1,1)\,\mathrm{nC}.$$

(b) Interpretation and further calculation

$$BC\text{ first}:\quad(12,0,-4)\longrightarrow(12,-2,-2).$$$$AB\text{ next}:\quad(12,-2,-2)\longrightarrow(5,5,-2)\,\mathrm{nC}.$$$$Q_{\mathrm{total}}=8\,\mathrm{nC}\quad\text{in either sequence}.$$Different final distributions can have the same total charge because the second connection acts on a different intermediate state.

Physical check

Equal sharing uses both conductor equilibration and symmetry; it is not a rule for arbitrary unequal objects.